The vapour pressure of pure liquid `M & N` are `700mm` of `Hg` and `450mm` of `Hg` respectively. Which of the following option is correct ?
Given : `X_(N),X_(M)=` mole fraction of `N & M` in liquid phase
`Y_(N),Y_(M)=` mole fraction of `N & M` in vapour phase

Options

A. `X_(M)-X_(N)gtY_(M)-Y_(N)`
B. `(X_(M))/(X_(N))gt(Y_(M))/(Y_(N))`
C. `(X_(M))/(X_(N))lt(Y_(M))/(Y_(N))`
D. `(X_(M))/(X_(N))-(Y_(M))/(Y_(N))`

Answer

To solve the problem, we need to analyze the given vapor pressures of the pure liquids M and N, and how they relate to their mole fractions in both the liquid and vapor phases.

### Step 1: Identify the given data
- Vapor pressure of pure liquid M (P₀M) = 700 mmHg
- Vapor pressure of pure liquid N (P₀N) = 450 mmHg

### Step 2: Determine the volatility of the liquids
Since the vapor pressure of liquid M is greater than that of liquid N (P₀M > P₀N), we can conclude that liquid M is more volatile than liquid N. This means that at equilibrium, more molecules of M will escape into the vapor phase compared to N.

### Step 3: Define the mole fractions
Let:
- Xₘ = mole fraction of M in the liquid phase
- Xₙ = mole fraction of N in the liquid phase
- Yₘ = mole fraction of M in the vapor phase
- Yₙ = mole fraction of N in the vapor phase

### Step 4: Apply Raoult's Law
According to Raoult's Law, the partial vapor pressure of each component in a mixture is given by:
- Pₘ = Xₘ * P₀M
- Pₙ = Xₙ * P₀N

The total vapor pressure (P_total) of the mixture can be expressed as:
\[ P_{total} = Pₘ + Pₙ = Xₘ * P₀M + Xₙ * P₀N \]

### Step 5: Analyze the mole fractions in the vapor phase
Since M is more volatile, we expect that:
- Yₘ > Yₙ

This indicates that the mole fraction of the more volatile component (M) in the vapor phase is greater than that of the less volatile component (N).

### Step 6: Compare the ratios of mole fractions
From the above observations, we can derive the relationship between the mole fractions in the liquid and vapor phases:
- Since Yₘ > Yₙ and M is more volatile, we can infer that:
\[ \frac{Xₘ}{Xₙ} < \frac{Yₘ}{Yₙ} \]

### Conclusion
Thus, the correct option from the given choices is:
\[ \frac{Xₘ}{Xₙ} < \frac{Yₘ}{Yₙ} \]